Elliptic curves over $\mathbb{F}_p$ and determinants of Legendre matrices

TL;DR

This work investigates determinants with Legendre-symbol entries and their deep ties to character sums and elliptic curves over finite fields. By combining eigenvalue decompositions, Zolotarev's lemma, and supersingular-elliptic-curve criteria, it confirms Sun's conjectures and derives precise evaluations for a broad family of determinants, including $(c,d)_p$, $[c,d]_p$, $W_p$, and $Y_p$, as well as the two-variable determinant $T_p$. The paper also connects these determinants to central combinatorial identities (via Grinberg–Sun–Zhao) and to arithmetic invariants such as $a_p(c)$, yielding infinite families of primes with prescribed determinant behavior. Collectively, the results illuminate the interplay between quadratic characters, higher-degree residue structures, and the arithmetic of elliptic curves, with implications for character sums and finite-field determinants.

Abstract

Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun and investigate some related topics. For instance, given any integers $c,d$ with $d\ne0$ and $c^2-4d\ne0$, we show that there are infinitely many odd primes $p$ such that $$\det\bigg[\left(\frac{i^2+cij+dj^2}{p}\right)\bigg]_{0\le i,j\le p-1}=0,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.

Theorems & Definitions (21)

• Conjecture 1.1
• Theorem 1.1
• Conjecture 1.2
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Conjecture 1.3
• Corollary 1.1
• Theorem 1.4
• Theorem 1.5